Wednesday 9 July 2014 03.00 EDT
First published on Wednesday 9 July 2014 03.00 EDT

A 1940s New York train carriage sits at rest inside a darkened warehouse, one of its sides cracked open. Ten actors in fedoras and tea dresses scuttle through it, followed by a camera crew. What they film is projected live on to a screen up above. "Camera – go," a woman whispers repeatedly into her headset, while two dramatists hiss bilingual instructions to the Anglo-German cast. Meanwhile, a young woman is frantically Googling German men's names on her laptop. "We can't use Fritz twice," she says. "Can we?"

In the middle of these rehearsals, taking place not far from London's Olympic stadium, sits Katie Mitchell in a black shirt dress, arms folded, back very straight. Mitchell is halfway through five weeks of rehearsals for Forbidden Zone, a play that will premiere at the Salzburg festival in August – though rehearsing is perhaps not quite the right word. Playwright Duncan Macmillan is writing it on set, and most of the time will be spent not so much refining the action as mapping out what is actually possible. "The danger with video is that when one bit goes wrong, the whole structure can collapse," says Mitchell. "Sometimes shows just stop."

The play – based on the story of the chemist Clara Immerwahr, who killed herself after the poison gas invented by her husband, Fritz Haber, was used for mass murder in the first world war – is the most ambitious yet by a director famous for daring productions. The initial idea was to have a train carriage, then Mitchell decided it had to look like a real train carriage from 1940s New York. Next she decided the train should open up to reveal an authentic replica of the Habers' 1913 Berlin flat. And could the toilet cubicle in the train station turn into a gas chamber?

"As the ambition of Katie's storytelling has expanded, we've needed to expand the canvas," says Leo Warner, the video director who has worked with Mitchell since 2007. As the project has grown in scale, so has the risk of it falling to pieces on the night. "We have absolute precision stage management", he sighs, "but this is complicated even by our standards."

Mitchell, who turns 50 this September, has been hailed as the closest thing British theatre has to a genuine auteur: a director with a strong, uncompromising vision of how theatre should be. But over the past four years, she has quietly slipped off the UK's radar. She first emerged as a director who did things differently with her 2007 multimedia adaptation of Virginia Woolf's The Waves. This was the beginning of what she now calls "live cinema": performances that come alive somewhere between the chaotic scramble on stage and the smooth, cinema-quality output on the screen.

'A new theatrical language' … Katie Mitchell's production of Beckett’s Footfalls in Berlin. Photograph: Christian Jungeblodt for the Guardian

Asked about those years now, she is careful not to seem bitter: "It may be that the work I make is at times too 'other' and so won't always fit tidily into mainstream culture in the UK. That sense of other could be to do with gender, or intellectualism, or wanting to experiment with form, or simply a feeling of Europeanism. I don't know."

Yet on the continent, her star has been rising. In the past two years, she has put on six productions at Germany and Austria's leading theatres, and is now rehearsing a string of new shows. "Over here, she's right up there," says Yvonne Büdenhölzer, director of the Berlin Theatertreffen festival, where Mitchell was this year's guest curator. Mitchell regularly commutes (always by train, never by plane) between London, where her young daughter lives, and Berlin, where several of her plays are part of the repertoire at the Schaubühne, where she has been allowed to experiment more freely.

Last December's production of Lungs by Macmillan, who has known Mitchell ever since he appeared in a student production of hers aged eight, told the story of a young couple worrying about having a baby in a era of global warming and overpopulation. Mitchell put the two actors on exercise bikes, wired up to produce the electricity that lights the stage: when they stop pedalling at the end of the play, the auditorium falls into darkness.

In last year's production of Charlotte Perkins Gilman's The Yellow Wallpaper, she built a film studio in miniature: a sound stage, a cinema screen, a booth in which a foley artist created live sound effects, and a cabin in which an actor voiced the main character's fragmented thoughts. One traditional criticism of Mitchell's approach is that she imposes her trademark style regardless of the original material's intentions. Yet the German press hailed this show as the apotheosis of her approach: the cinematic surveillance brought out the Gothic paranoia of Gilman's 19th-century short story. "A new theatrical language," critic Sascha Krieger called it, "as intense and eye-opening as only theatre can be."

One explanation for Mitchell's continental success is that she has always aspired to the German ideal of Regietheater, which prioritises the director's interpretation over the writer's intention. "In Germany, we aren't thrown by directors who are irreverent towards the original material," Peter Laudenbach, theatre critic for Süddeutsche Zeitung, says. "It's commonplace now."

We meet again in June, at Berlin's Schiller Theater, during rehearsals for a double performance of Samuel Beckett's Footfalls and Morten Feldman's Either. "Even though I was in the mainstream of British theatre for over 20 years," says Mitchell, "I am suddenly here. Is that because my influences unconsciously were always from other countries? That's what I ask myself."

Although she cites choreographer Pina Bausch and eastern European directors including Anatoly Vasiliev and Lev Dodin as influences, there are still some very British qualities at the heart of Mitchell's theatre. One of them is "a very British love of good craft", according to Laudenbach. "German directors like their work expressive, sweaty but authentic. And German critics are always instinctively sceptical when a play looks too polished – they fear it may cover up a lack of depth."

The 1940s New York train carriage that forms part of the set for Katie Mitchell's new show The Forbidden Zone. Photograph: Stephen Cummiskey

Mitchell, whose productions speak to both traditions, insists the films in her plays should look like cinema – hence the detailed rehearsals. "I never think: 'Now it clicks.' I never think: 'This is my style.' I just think, every 10 seconds: 'Is that good enough or not?' There's no sense of arrival or achievement, no armchair at the end of everything; just another challenge."

Although Mitchell has a reputation for being fierce and for over-intellectualising, she is not like that in person: warm, chatty, with a very British tendency to make light of her more abstract ideas. One minute she is theorising earnestly about the difference between Germany and Britain's postwar traumas, the next she is giggling as if she's just caught herself auditioning for Pseuds Corner: "I wonder what status being the victor gives to the trauma. Doesn't it slightly marginalise it?" Then a loud laugh, and the words: "Ah well, la di da."

Unlike most directors working in Germany, Mitchell expects her actors to research or imagine the biographies of their characters – another aspect closer to the realist rather than the Brechtian tradition. "She's trying to examine people in increasingly forensic detail," Leo Warner says. "She fragments them then brings them back together."

Forbidden Zone's rehearsals suggests a director not so much running away from British theatre as trying to rethink its traditions – which makes it seem slightly absurd that fans of her work now have to travel to Germany to see it. Will she be remembered as another Peter Brook, who made most of his significant contributions to British culture while working abroad?

She replies by wondering if she will be remembered at all. "How long can anything you made last?" she says. "I am not very optimistic about that." She recalls seeing Welcome to Sarajevo, Michael Winterbottom's film about the war in Bosnia. "It made me aware of the fragility of our existence. I realised war can just come and annihilate all of us. On the tube home afterwards, I tried really hard to find a seat for an elderly woman so she wouldn't bump into people. But two days later, the feeling had gone."

• The Forbidden Zone will be live-streamed from the Salzburg festival on 2 August at the Barbican, London EC2.

• This article was amended on 9 July 2014. An earlier version referred to a take on Purcell's Dido and Aeneas at the National. That should have been After Dido, a collaboration with English National Opera and the Young Vic, staged at the Young Vic.